MATHEMATICS

I
MEAN VALUE THEOREMS
FUNCTIONS OF SINGLE
&
SEVERAL VARIABLES
I YEAR B.TECH
By
Y. Prabhaker Reddy
Asst. Professor of Mathematics
Guru Nanak Engineering College
Ibrahimpatnam, Hyderabad.
SYLLABUS OF MATHEMATICS

I
(
AS PER JNTU HYD
)
Name of the Unit
Name of the Topic
Unit

I
Sequences and Series
Basic definition of sequences and series
Convergence and divergence.
Ratio test
Comparison test
Integral test
Cauchy’s root test
Raabe’s test
Absolute and conditional convergence
Unit

II
Functions of single variable
Rolle’s theorem
Lagrange’s Mean value theorem
Cauchy’s Mean value theorem
Generalized mean value theorems
Functions of several variables
Functional dependence, Jacobian
Maxima and minima of function of two variables
Unit

III
Application of single variables
Radius , centre and Circle of curvature
Evolutes and Envelopes
Curve Tracing

Cartesian Co

ordinates
Curve Tracing

Polar Co

ordinates
Cu
rve Tracing

Parametric Curves
Unit

IV
Integration and its
applications
Riemann Sum
Integral representation for lengths
Integral representation for Areas
Integral representation for Volumes
Surface areas in Cartesian and Polar co

ordinates
Multiple
integrals

double and triple
Change of order of integration
Change of variable
Unit

V
Differential equations of first
order and their applications
Overview of differential equations
Exact and non exact differential equations
Linear differential equations
Bernoulli D.E
Newton’s Law of cooling
Law of Natural growth and decay
Orthogonal trajectories and applications
Unit

VI
Higher order Linear D.E and
their applications
Linear D.E of second and higher order with constant coefficients
R.H.S term of the form
e
xp(ax)
R.H.S term of the form
sin ax and cos ax
R.H.S term of the form exp(ax) v(x)
R.H.S term of the form exp(ax) v(x)
Method of variation of parameters
Applications on bending of beams, Electrical circuits and simple harmonic motion
Unit

VII
Laplace
Transformations
LT of standard functions
Inverse LT
–
first shifting property
Transformations of derivatives and integrals
Unit step function, Second shifting theorem
Convolution theorem

periodic function
Differentiation and integration of transforms
Applic
ation of laplace transforms to ODE
Unit

VIII
Vector Calculus
Gradient, Divergence, curl
Laplacian and second order operators
Line, surface , volume integrals
Green’s Theorem and applications
Gauss Divergence Theorem and applications
Stoke’s Theorem and
applications
CONTENTS
UNIT

2
Functions of Single & Several Variables
Rolle’s Theorem(without Proof)
Lagrange’s Mean Value Theorem(without Proof)
Cauchy’s Mean Value Theorem(without Proof)
Generalized Mean Value Theorem (without Proof)
Functions of
Several Variables

Functional dependence
Jacobian
Maxima and Minima of functions of two variables
Introduction
Real valued function
: Any function
is called a Real valued function.
Limit of a function:
Let
is a real valued function and
. Then, a real number
is
said to be limit of
at
if for each
whenever
. It is denoted by
.
Continuity:
Let
is a real valued function and
. Then,
is said to be continuous at
if for each
whenever
. It is
denoted by
.
Note:
1) The function
is continuous every where
2) Every
function is continuous every where.
3) Every
function is not continuous, but in
the function
is continuous.
4) Every polynomial is continuous every where.
5) Every exponential function is continuous every where.
6) Every lo
g function is continuous every where.
Differentiability:
Let
be a function, then
is said to be differentiable or derivable at
a point
, if
exists.
Functions
Algebraic
Trignonometric
Inverse
Exponential
Logarithmic
Hyperboli
c
MEAN VALUE THEOREMS
Let
be a given function.
ROLLE’S THEOREM
Let
such that
(i)
is continuous in
(ii)
is differentiable (or) derivable in
(iii)
then
atleast one point
in
such that
Geometrical Interpretation of Rolle’s Theorem:
From the diagram, it can be observed that
(i) there is no gap for the curve
from
. Therefore, the function is
continuous
(ii) There exists unique tangent for every intermediate point between
and
(ii) Also the ordinates of
and
are same, then by Rolle’s theorem, there exists atleast one point
in between
and
such that tangent at
is parallel to
axis.
LAGRANGE’S MEAN VALUE THEOREM
Let
such that
(i)
is continuous in
(ii)
is differentiable (or) derivable in
then
atleast one point
such that
Geometrical Interpretation of Rolle’s Theorem:
From
the diagram, it can be observed that
(i) there is no gap for the curve
from
. Therefore, the function is
continuous
(ii) There exists unique tangent for every intermediate point between
and
T
hen by
Lagrange
’s
mean value
theorem, there exists atleast one point
in between
and
such that tangent at
is parallel to
a straight line joining the points
and
CAUCHY’S
MEAN VALUE THEOREM
Let
are such that
(i)
are
continuous in
(ii)
are
differentiable (or) derivable in
(iii)
then
atleast one point
such that
Generalised
Mean Value Theorems
Taylor’s Theorem:
If
such that
(i)
are continuous on
(ii)
are derivable or differentiable on
(iii)
, then
such that
where,
is called as Schlomilch
–
Roche’s form of remainder and is given by
Lagrange’s form of Remainder:
Substituting
in
we get Lagrange’s form of Remainder
i.e.
Cauchy’s form of Remainder:
Substituting
in
we get Cauchy’s form of Remainder
i.e.
Maclaurin’s Theorem:
If
such that
(i)
are continuous on
(ii)
are derivable or differentiable on
(iii)
, then
such that
where,
is called as Schlomilch
–
Roche’s form of remainder and is given by
Lagrange’s form of Remainder:
Substituting
in
we get Lagrange’s form of Remainder
i.e.
Cauchy’s form of Remainder:
Substituting
in
we get Cauchy’s form of Remainder
i.e.
Jacobian
Let
are two functions , then the Jacobian of
w.r.t
is
denoted by
and is defined as
Properties:
If
are functions of
and
are functions of
then
Functional Dependence:
Two functions
are said to be functional dependent on one another if the Jacobian of
is zero.
If they are functionally dependent on one another, then it is possible to find the relation between
these two functions.
MAXIMA AND MINIMA
Maxima and Minim
a for the function of one Variable:
Let us consider a function
To find the Maxima and Minima, the following procedure must be followed:
Step 1:
First find the first derivative and equate to zero. i.e.
Step 2:
Since
is a polynomial
is a polynomial equation. By solving this
equation we get roots.
Step 3:
Find second derivative i.e.
Step 4:
Now substitute the obtained roots in
Step 5:
Depending on the Nature of
at that point we will solve further. The following cases will
be there.
Case (i):
If
at a point say
, then
has maximum at
and the maximum
value is given by
Case (ii):
If
at a point say
, then
has minimum at
and the minimum
value is given by
Case (iii):
If
at a point say
, then
has neither minimum nor maximum. i.e.
stationary.
Maxima and Minima for the function of Two Variable
Let us consider a funct
ion
To find the Maxima and Minima for the given function, the following procedure must be followed:
Step 1:
First find the first derivatives and equate to zero. i.e.
(
Here, since we have two variables, we go fo
r partial derivatives, but not ordinary
derivatives
)
Step 2:
By solving
, we get the different values of
and
.
Write these values as set of ordered pairs. i.e.
Step 3:
Now, find second order partial derivatives.
i.e.,
Step 4:
Let us consider
,
Step 5:
Now, we have to see for what values of
, the given function is maximum/minimum/
does not have extreme values/
fails to have maximum or minimum.
If at a point, say
, then
has
maximum
at this point and the
maximum value will be obtained by substituting
in the given function.
If at a point, say
, then
has
minimum
at this point and the
m
in
imum value will be obtained by substituting
in the given function.
If at a point, say
, then
has
neither maximum nor minimum
and
such points are called as saddle points
.
If at a p
oint, say
, then
fails to have
maximum
or minimum and case
needs further investigation to decide maxima/minima
.
i.e.
No conclusion
Problem
1)
Examine the function for extreme values
Sol:
Given
The first order partial derivatives of
are given by
and
Now, equating first order partial derivatives to zero, we get
Solving
we get
Substituting
in
Substituting
in
All possible set of values are
Now, the second order partial derivatives are given by
Now,
At a point
&
has maximum at
and the maximum value will be obtained by substituting
in the
function
i.e.
is the maximum value.
Also, at a point
&
has minimum at
and the minimum value is
.
Al
so, at a point
has neither minimum nor maximum at this point.
Again, at a point
has neither minimum nor maximum at this point.
Lagrange’s Method of Undetermined Multipliers
This method is useful to find the
extreme values (i.e., maximum and minimum) for the given
function, whenever some condition is given involving the variables.
To find the Maxima and Minima for the given function using Lagrange’s Method , the following
procedure must be followed:
Step 1:
Le
t us consider given function to be
subject to the condition
Step 2:
Let us define a Lagrangean function
, where
is called the Lagrange
multiplier.
Step 3:
Find first order partial derivatives and equate to zero
i.e.
Let the given condition be
Step 4
:
Solve
, eliminate
to get the values of
Step 5:
The values so obtained will give the stationary point of
Step 6:
The minimum/maximum value will be obtained by substituting the values of
in the
given function.
Problem
1) Find the minimum value of
su
bject to the condition
Sol:
Let us consider given function to be
and
Let us define Lagrangean function
, where
is called the Lagrange multiplier.
Now,
Solving
Now, consider
Again, consider
Again solving
Given
Similarly, we get
Hence, the minimum
value of the function is given by
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